Exam Revision

Question 1

What are the subring criteria

Answer 1

Question 2

Give the definition of an integral domain

Answer 2

Question 3

Give the definition of a unit

Answer 3

Question 4

Give the definition of irreducible

Answer 4

Question 5

State Gauss’ Lemma

Answer 5

Question 6

State Eisenstein’s Criterion

Answer 6

Question 7

Give the definition of prime

Answer 7

Question 8

How to prove a homomorphism is a injective

Answer 8

Question 9

Give the criteria for ideals matching

Answer 9

Question 10

Give the definition of prime and maximal ideals

Answer 10

Question 11

What rings are PIDs

Answer 11

Question 12

What is the CRT for rings

Answer 12

Question 13

State Lagrange’s Theorem

Answer 13

Question 14

State the normality criteria

Answer 14

Question 15

Give the definition of conjugacy

Answer 15

Question 16

Define the kernel of a homomorphism

Answer 16

Question 17

A group of order p is isomorphic to

Answer 17

Question 18

What do isomorphisms preserve

Answer 18

Question 19

Give the criteria for subgroups to isomorphic to the parent group

Answer 19

Question 20

State Cayley’s Theorem

Answer 20

Question 21

State how orbits partition the set

Answer 21

Question 22

State the orbit stabliser theorem

Answer 22

Question 23

State the corollary of the OS theorem

Answer 23

Question 24

State Cauchy’s Theorem

Answer 24

Question 25

Groups of order 2p classification theorem

Answer 25

Question 26

What the conjugacy classes in Sn

Answer 26

Question 27

What are conjugacy classes in An

Answer 27

Question 28

What is the relationship between normality and ccls

Answer 28

Question 29

When is the centre non trivial

Answer 29

Question 30

Give the p^2 classification theorem

Answer 30

Question 31

State Sylow's Theorem

Answer 31

Question 32

State the fundamental theorem of finitely generated groups

Answer 32

Question 33

Give the formula for number of cycles of shape [m1, m2, .., mr]

Answer 33

Question 34

Give the formula for cycles of shape [m1,...,m1,m2..,m2...mr..mr]

Answer 34